I have the following 3 problems, where u is a $n \times 1$ column vector and $A,B$ are $n \times n$ matrices. Furthermore, all entries are complex numbers.
a. Show that if $A$ is Hermitian, $u^{*}Au$ is real.
b. Show that $B^{*}B$ is Hermitian.
c. Prove that $u^{*}B^{*}Bu$ is real and non-negative.
I have proven parts a and b. Note that by combining them, we may almost get c (the realness follows, but not the non-negativity).
After starting c from scratch and trying to prove it from the definition of matrix multiplication, I haven't gotten anywhere. I'm wondering whether there is a simpler approach to part c that doesn't throw away all the work I've already done.
We have $x^*x=\sum_j\bar{x_j}x_j=\sum_j|x_j|^2\ge0$ for any vector $x$. Apply it for $x=Bu$.
Note that $\langle x, y\rangle:=x^*y$ is an inner product, the above can be rewritten $\langle x, x\rangle\ge0$ (which is an axiom for inner products, and it induces a norm $\|x\|:=\sqrt{\langle x, x\rangle}$), so that we have $$u^*B^*Bu=\langle Bu, Bu\rangle=\|Bu\|^2\ge0$$